MODEL

In this model we consider a competitive two-sector economy in which one sector produces a final good, c₁, that is used either for consumption or for adding to the existing capital stock, while the other sector produces a final good, c₂, that is used for consumption only. Both sectors produce outputs by employing capital, k, and labor, l, with constant-returns-to-scale technology. Capital does not depreciate.

The production technologies are summarized as follows:

where F and G are constant-returns-to-scale production functions with standard properties. In the expression, we have normalized the total labor endowment to unity, so that l and (1−l) are the shares of labor allocated to the first and the second sectors, respectively. Terms z and (k−z) are the stocks of capital allocated to the first and the second sector, respectively. Let

where k_x≡z/l is the capital/labor ratio in the first sector and k_y≡(k−z)/(1−l) is the ratio in the second sector. The labor productivity functions, f and g, are assumed to be strictly increasing and strictly concave.

Given (1) and (2), it is clear that c₁ and

must sell for the same price in a competitive market, but c₁ and c₂ need not do so. Labor and capital are assumed to be mobile across industries. The structure of production in this paper can be thought of as a generalization of Huo (1997). The economy consists of an aggregation of identical, infinitely lived representative households, each of which maximizes an intertemporal utility function that is separable in c₁ and c₂:²

We solve here an optimization problem under competitive equilibrium, in other words, a social planning problem. The planner takes the technology constraints as given and seeks to maximize household utility over consumption plans. This means that the planner tries to maximize (3) subject to (1) and (2), where ρ>0 is a constant subjective rate of time preference. The instantaneous utility functions u₁(·) and u₂(·) are strictly increasing, strictly concave, continuously differentiable, and satisfy the Inada conditions. The representative household holds either capital (k) or cash (M). Cash is injected into the system through lump-sum transfers (and withdrawn by lump-sum taxes).

The household's budget constraint is

and the capital accumulation constraint is given by

with k(0) and M(0) given. In the expressions, p₁(t) and p₂(t) are the nominal prices of c₁ and c₂, respectively, at time t. Term T(t) is the lump-sum nominal transfer from the government at time t. It is assumed that the purchases of c₁ and c₂ are restricted by a CIA constraint of the following general form:

where ω₁ and ω₂ are, respectively, the fractions of cash needed to purchase c₁ and c₂, 0[les ]ω₁, ω₂[les ]1, ω₁+ω₂≠0∀t, and M(t), the cash balance at time t, is introduced to the system through the above CIA constraint.

Denoting as H the Hamiltonian of the optimization problem, α and β are the costate variables associated with M and k, respectively, λ is the Lagrangian multiplier attached to the CIA constraint, and i is the slack variable. We can therefore write H as

Applying Pontryagin's Maximum Principle, we find that the optimizing program is described by the following first-order conditions:

together with two transversality conditions,

The necessary conditions (7), together with Eqs. (1) and (2), can be used to characterize the solution to (3). A sufficient condition for a solution to (3) to exist is that the Hamiltonian be concave in (l, k), which is met, given the specification of preferences and technologies. Equation (7a) [(7b)] equates the marginal utility of c₁(c₂) per dollar spent to the sum of the marginal utility of income and the fractions of the purchase of c₁(c₂) constrained by cash times the marginal utility of cash. Equation (7c) [(7d)] equates the marginal products of capital (labor) in each sector. Equation (7e) equates the marginal benefit of investment per dollar spent to the marginal benefit of income. Equation (7f) [(7g)] describes the dynamic behavior of the marginal benefit of income (investment). Equation (7h) [(7i)] rules out Ponzi-game behavior in trading capital (cash).

We define p=p₂/p₁ to be the relative price of the two consumption goods. Equations (7c) and (7d) require that the real rates of return on capital and labor be equalized across sectors. Let w≡f(k_x)−k_xf′(k_x) denote the market real wage on labor and r≡f′(k_x) denote the market real rental on capital. Equations (7c) and (7d) can be used to solve for the factor intensities in each sector and factor prices as functions of p alone. Totally differentiating (7c) and (7d) and using the definition of factor prices will yield

By (8c) and (8d), it can be seen that there is a monotonic relation between the relative costs (prices) of the goods and the relative factor prices, which is the well-known Stolper–Samuelson relation.³

The relation is a consequence of the assumed difference in factor intensities between commodities, and holds so long as both goods are produced.⁴

The solutions to (8) can be combined with the full employment condition, which requires that lk_x(p)+(1−l)k_y(p)=k, which implies that

Equation (9) shows that the optimum labor fraction is a function of the relative price and capital stock.

STEADY-STATE ANALYSIS

The equilibrium conditions for the economy require that the goods markets, factor markets, and money market all clear. Equilibrium in the goods market is represented above by (1) and (2), whereas factor market equilibrium requires that both labor and capital be fully employed. The money market equilibrium implies a binding CIA constraint.

To simplify the system of (7), we first substitute (7d) and (7e) into (7g) to obtain

By (7a) and (7b), we next obtain

Taking logarithms of both sides of the above equation and differentiating with respect to time results in

By (7a), (7b), and (7e), we can solve for λ, and then substituting the value of λ into (7f), we obtain

The money market clearing condition implies that the CIA constraint is binding. Taking logarithms of both sides of (6) and differentiating with respect to time,

The money supply is assumed to follow a constant rate of growth, μ,

The condition of μ[ges ]−ρ is generally known to ensure the existence of a monetary steady state. Substituting (10), (13), and (15) into (14), we can solve for

. The second-good market-clearing condition implies that we can substitute (2) into (12) to obtain

where

Note that in (16), l≡l(p, k), f≡f[k_x(p)], and g≡g[k_y(p)]; therefore,

is a function of c₁, k, and β. The equilibrium motion of (c₁, k, β) is thus completely characterized by (1), (10), and (16).⁵

In the steady state,

. Before the formal comparative analysis, it should be noted that equation (10) implies that the steady-state marginal productivity of capital is tied to the rate of time preference:

Equation (17) shows that the bar over the variable denotes its steady-state value, implying that the steady-state marginal productivity of capital is tied to the rate of time preference. The steady-state real rate of return on capital is determined by the Modified Golden Rule (without population growth) from the traditional optimal growth theory. Here, we can interpret the relationship between the relative price of commodities and the rate of time preference. On the one hand, we observe a monotonic relation between the relative prices of the goods (p) and the relative factor prices (r, w) in equation (8) as Stolper–Samuelson predicted. On the other hand, r≡f′[k_x(p)]=ρ in the steady state, which represents that the rental price of capital is equal to a constant rate of time preference.

It therefore follows that the steady-state relative commodity price is determined by an exogenous rate of time preference, and therefore is irrelevant to the inflation rate. The result is surprising compared to the other two-sector models [Huo (1997), Mino (1997)]. The constancy of the relative commodity price comes from the CIA constraint. In this model, we assume only that the purchases of consumption goods need cash, whereas the purchases of investment and factor inputs (labor and capital) do not. Since cash is not needed to purchase factor inputs, inflation has no influence on the relative factor price. In addition, it is known that there is a monotonic relation between the relative factor price and the relative commodity price; therefore, it is trivial to see that the relative commodity price is also independent of the inflation rate.

We may now conjecture that when investment purchases are also subject to the CIA constraint, the proposition that inflation does not affect the relative commodity price will no longer be valid.⁶

By exploiting the CIA setting in the Heckscher–Ohlin model, this paper actually provides a mechanism to bring the well-known characteristics (i.e., Stolper–Samuelson relation, golden rule) of a real model to a model with money.

To elucidate the relationship between anticipated inflation and the steady-state capital stock, we rewrite (14) as

Since

in the steady state and (17) implies that

, it follows that

, and

. We substitute (13) and (15) into (18) to obtain

From (7a), (7b), and (7e), we see that in the steady state,

To investigate the effects of permanent anticipated inflation on the steady-state capital stock, we first differentiate (19) and (20). From the goods-markets equilibrium conditions of (1), (2), and (7), it is known that in the steady state,

and

. By applying the relations that

and

, we have

This in turn implies that

where l_k=1/(k_x−k_y). Since p>0, β>0, f, g>0, u″₁, u″₂<0, equation (21) shows that the factor intensities (k_x and k_y) and the fractions of consumption purchases made using cash (ω₁ and ω₂) determine the effects of inflation on steady-state capital accumulation. The results can be shown as follows:

The results show that factor intensities and the fractions of purchases of consumption goods constrained by CIA determine the effect of permanent anticipated inflation on the steady-state capital stock. Moreover, if a more capital-intensive good is also less (more) cash intensive, then inflation will induce higher (lower) capital accumulation. The reasoning for the results can be illustrated by supposing that the economy is in a steady state initially.

1. We first consider the case where the product is relatively capital intensive and less cash intensive [(k_x>k_y and ω₁<ω₂) or (k_x<k_y and ω₁>ω₂)]. Higher inflation raises the opportunity cost of purchasing a cash-intensive good; this induces the utility-maximizing households to make a new consumption choice on their indifference curves, and results in a substitution of credit good for cash good. Since the less cash-intensive good is relatively more capital-intensive, the increase in the demand for the less cash-intensive good leads to more capital accumulation.
2. It is easy to see that, vice versa, higher inflation induces capital decumulation when a sector is labor-intensive and less cash-intensive [(k_x<k_y and ω₁<ω₂) or (k_x>k_y and ω₁>ω₂)].
3. It is also clear that, regardless of the factor intensity, as long as the purchases of the two consumption goods are equally constrained by the CIA (ω₁=ω₂), then higher inflation equally affects the purchases of the two goods. It does not lead to any distributional effect in the production of the two sectors; therefore, higher inflation has no effect on capital accumulation.

We generally observe reversed Tobin effects when the capital-intensive good is also a cash-intensive good and Tobin effects when the capital-intensive good is a less cash-intensive good. The steady-state effect of inflation on capital accumulation comes solely from the substitution effect between the two consumption goods made by inflation, not from the changes in the relative prices. Although this model does not take capital depreciation into consideration, it is found that adding depreciation to the model will not alter the results if the depreciation is relatively small.⁷

CIA MODEL WITH ENDOGENOUS LABOR SUPPLY

The aim of this section is to see through the role of endogenous labor supply in this model. The model is similar to the model in the preceding section except the production technologies are modified as follows:

where l₁ and l₂ are the shares of labor allocated to the first and the second sector, respectively. We next let

where k_x≡z/l₁ is the capital/labor ratio in the first sector, and k_y≡(k−z)/l₂ is the ratio in the second sector.

The economy consists of an aggregation of identical, infinitely lived representative households each of which maximizes an intertemporal utility function that is separable in c₁ and c₂ and leisure [ell ]:

where ρ>0 is a constant subjective rate of time preference. The instantaneous utility functions u₁(·), u₂(·), and ν(·) are all strictly increasing, strictly concave, continuously differentiable, and satisfy the Inada conditions. We normalize the time endowment to unity, which is l₁+l₂+[ell ]=1. The Hamiltonian H is

The first-order conditions are the same as in equation (7) except that there is an extra FOC in this section as the following:

We use the same methodology to simplify the FOC and market-clearing conditions. By applying the relations of

and

, we obtain

and

where

.

It is quite trivial that, from (22), inflation leads to higher demand for leisure by reducing the opportunity cost of leisure. Hence, higher inflation reduces the total labor supply with no doubt in the model concerning the endogenous labor/leisure choice. This result is consistent with that of Brock (1974), but different from that of Yip (1991).⁸

Note that

, and σ=−v′/[ell ]v″,⁹

which represents the elasticity of the intertemporal substitution of leisure. Since p>0, β>0, f, g>0, and u″₁, u″₂, v″<0, equation (21″) shows that aside from the factor intensities (k_x and k_y) and the fractions of consumption purchases made by using cash (ω₁ and ω₂), the elasticity of intertemporal substitution of leisure also plays a role in determining the effects of inflation on steady-state capital accumulation:

The steady-state effect of inflation comes mainly from factor redistribution in the Heckscher–Ohlin model. Comparing the results with last section's, we definitely find a negative role of endogenous labor supply in the real effects of money growth, which is in accordance with Brock (1974), Stockman (1985), Yip (1991), Gomme (1993), and Mino (1997). A sector that produces a capital-intensive and less cash-intensive good is no longer guaranteed to induce higher capital accumulation for an increase in inflation rate. Here, we should be more careful in presenting the redistribution effect of labor input.

Note that

, and in order for us to clarify the redistribution effect of labor supply by inflation between the two sectors, we obtain

,

in most cases; whereas

only occurs when ω₁<ω₂ and

; and

only occurs when ω₁>ω₂ and σ<[(ω₂−ω₁)v′]/([ell ]wω₂fu″₁)≡σ_B. We further obtain the relations that

,

. This shows that inflation affects steady states c₁ and c₂ in the same direction as it affects l₁ and l₂, respectively.

For the effects of inflation on capital accumulation, we obtain a reversed Tobin effect in most cases when the endogenous labor/leisure choice is taken into account. The economic reasoning is as follows:

1. We first consider the case that a capital-intensive sector is also cash-intensive [(k_x>k_y and ω₁>ω₂) or (k_x<k_y and ω₁<ω₂)]. The income effect of inflation leads to a decrease in the purchases of both goods. In addition, the higher inflation decreases the demand for the cash-intensive good and therefore leads to less capital accumulation [(dk/dμ)<0] since the cash-intensive good is a capital-intensive good. Total labor supply (l₁+l₂) decreases because leisure becomes more valuable in a higher inflation period. The share of labor allocated to the cash-intensive sector decreases with certainty. In most cases, the shares of labor allocated to the less cash-intensive sector decrease and therefore the demand for the less cash-intensive goods decreases; only when the elasticity of intertemporal substitution of leisure is extremely small is the chance that the labor ratio and the demand for the goods in this sector can increase. Long-run capital stock definitely decreases with an increase in inflation.
2. In the case of ω₁=ω₂, inflation equally affects the opportunity costs of the two consumption purchases. The income effect of inflation leads to a decrease in the demand for both consumption goods. Inflation also equally affects the shares of factor input allocated to the two sectors, leading to a reduction in both l₁ and l₂. The decrease in both demand for consumption goods and labor supply of the two sectors causes the demand for capital to decrease, and therefore the long-run capital stock is reduced. We obtain a reversed Tobin effect when ω₁=ω₂. When comparing this with the superneutral result in the preceding section, the negative role of endogenous labor supply is apparent.
3. The steady-state effect of inflation on real economic variables is complex and ambiguous in the case where the capital-intensive good is also a less cash-intensive good [(k_x<k_y and ω₁>ω₂) or (k_x>k_y and ω₁<ω₂)]. Higher inflation results in a substitution of the less cash-intensive good for the cash-intensive good. The steady-state level of the cash-intensive good and the labor ratio of this sector definitely decrease, but the values in the other sector may decrease or increase, depending upon the magnitude of the elasticity of the intertemporal substitution of leisure. Since the cash-intensive good is labor-intensive and the less cash-intensive good is capital-intensive, the steady-state capital stock decreases unambiguously when the demand for both goods decreases; but it may increase, decrease, or not change when the demand for the less cash-intensive good increases. In contrast to the results in the exogenous labor supply model where inflation unambiguously leads to higher capital accumulation, the concern of the elasticity of intertemporal substitution of leisure plays an important role.

Although, in general, we obtained a Tobin effect (i.e., capital accumulation), a reversed Tobin effect (i.e., capital decumulation), or a zero effect again as in the preceding section, we do find a negative role of endogenous labor supply. This section requires further consideration on the elasticity of the intertemporal substitution of leisure and the redistribution between labor and leisure. Through the redistribution effects in factor inputs, we conclude that inflation has a positive effect on steady-state leisure and has negative effects on other variables (labor supply, consumption goods, and capital accumulation) in most cases.

We can also demonstrate the effect of inflation on economic welfare:

The sign of the above expression is uncertain. For simplicity, we assume that the condition is used to ensure that the existence of a monetary steady state (μ[ges ]−ρ) is binding. Replacing by μ+ρ=0, the above expression can be reduced to

We derive that sign (dU/dμ) = sign (

is negative in most cases and nonnegative in fewer cases. Accordingly, it implies that, in general, an increase in the rate of money growth reduces welfare in the steady state, which is consistent with the results in Bailey (1956), Lucas (1981), Cooley and Hansen (1989, 1991), and Yip (1991).¹⁰

Compared to other literature, this paper is more general and comprehensive by combining all situations in one model and being able to explain all situations.